Finite Group Databasebeta

← #29608

Subgroups — #29608

Order 256.

666 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#537C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4#2616C211yesmaximal, characteristic, complemented
5#2036C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#2000C212yesmaximal, complemented
9#1823C211yesmaximal, characteristic, complemented
10#1788C211yesmaximal, characteristic, complemented
11#1780C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C22×Q16C2211yescharacteristic
15C22×SD16C2211yescharacteristic
16#537—24no—
17#537—24no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2212yes—
21C2×C22⋊Q8C2212yes—
22C2×#121—21no—
23C2×#121—22no—
24C2×#121C2212yes—
25C22×C4⋊C4C2211yescharacteristic
26#484—24no—
27#452—24no—
28C2×#107C2212yes—
29C2×#103C2212yes—
30C2×#103C2211yescharacteristic
31C2×#103C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2212yes—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2212yes—
37C2×C22⋊C8C2212yes—
38#395—21no—
39#394—22no—
40#394—21no—
41#380C2211yescharacteristic
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—22no—
48C2×#41—42no—
49C2×#41—42no—
50C2×#41—42no—
51C2×#41—42no—
52C2×#41D414yes—
53C22×Q8D411yescharacteristic
54C22×Q8C2311yescharacteristic
55C22×Q8C2311yescharacteristic
56C22×D4—21no—
57C22×D4—22no—
58C22×D4D412yes—
59C22×D4D411yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4C2312yes—
64C23×C4C2311yescharacteristic
65C2×Q16—22no—
66C2×Q16—42no—
67C2×SD16—22no—
68C2×SD16—42no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2312yes—
72C22⋊Q8—28no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—42no—
78C22⋊Q8—42no—
79#121—216no—
80#121—28no—
81#121—24no—
82#121—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D414yes—
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2312yes—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—21no—
98#107—24no—
99#103—22no—
100#103—24no—
101#103—42no—
102#102—22no—
103#102—42no—
104C22⋊C8—22no—
105C22⋊C8—22no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#95—22no—
109#95—21no—
110C24—21no—
111C24C2×D412yes—
112C24C2×D411yescharacteristic
113#41—44no—
114#41—44no—
115#41—44no—
116#41—44no—
117#41—48no—
118#41—48no—
119C2×Q8—22no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—42no—
128C2×Q8C2×D412yes—
129C2×D4—24no—
130C2×D4—24no—
131C2×D4—28no—
132C2×D4—22no—
133C2×D4—21no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—44no—
140C2×D4—82no—
141C2×D4—82no—
142C2×D4C2×D412yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—44no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4C2×D412yes—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D412yes—
164C22×C4C2×D412yes—
165C22×C4C2411yescharacteristic
166Q16—44no—
167SD16—44no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C4⋊C4—28no—
176C4⋊C4—24no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—44no—
183C4⋊C4—42no—
184C4⋊C4C2×D414yes—
185C22⋊C4—28no—
186C22⋊C4—28no—
187C22⋊C4—44no—
188C22⋊C4—42no—
189C22⋊C4—81no—
190C22⋊C4—82no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—42no—
204C23—41no—
205C23—41no—
206C23—42no—
207C23#12012yes—
208C23C22×D411yescharacteristic
209Q8—22no—
210Q8—22no—
211Q8—44no—
212Q8—42no—
213Q8—41no—
214Q8—41no—
215Q8—42no—
216D4—22no—
217D4—22no—
218D4—44no—
219D4—48no—
220D4—48no—
221D4—42no—
222D4—88no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—48no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—84no—
249C2×C4#12012yes—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4H(5,𝔽2)12yes—
253C2×C4H(5,𝔽2)12yes—
254C8—41no—
255C8—41no—
256C8—42no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—82no—
273C22—82no—
274C22C2×#12011yescharacteristic
275C22#53412yes—
276C22#57611yescharacteristic
277C22#57711yescharacteristic
278C22#57812yes—
279C4—22no—
280C4—42no—
281C4—42no—
282C4—44no—
283C4—81no—
284C4—82no—
285C4—81no—
286C4#53411yescharacteristic
287C4#53411yescharacteristic
288C4#53512yes—
289C2—42no—
290C2—82no—
291C2—82no—
292C2#261311yescharacteristic
293C2#301111yescharacteristic
294C2#301211yescharacteristic
295C2#320112yes—
296C2#320312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#484C2semidirect
3. C2×#452C2semidirect
4. #2616C2semidirect
5. #2036C2semidirect
6. #2033C2semidirect
7. #2001C2semidirect
8. #2000C2semidirect
9. #1823C2semidirect
10. #1788C2semidirect
11. #1780C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#5372
C2×#4842
C2×#4522
#26162
#20362
#20332
#20012
#20002
#18232
#17882
#17802